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Full text of "Analytical Mechanics"

206
ANALYTICAL MECHANICS
ILLUSTRATIVE EXAMPLES.
1. Find the force which a particle placed upon a smooth inclined plane experiences by virtue of its potential energy. Also find the components of the force along the axes of a rectangular system, in which the z-axis is normal to the inclined plane and the o>axis is horizontal.
Let the origin of the axes, Fig. 104, be the position of the particle. Then if h denotes its height from the base of the inclined plane the potential energy is mgh. Therefore the force along the vertical is given by
FIG. 104.
Thus the force due to the gravitational field is downwards and equals the weight of the particle. The components of the force are found by equation (I7). Thus
dU
dx
v
X =   = -
dh     n  - = 0.
dx
Therefore the force along the re-axis is nil.
v         dU            dh
Y  -- =  ma  =  mg sin a.. dij             * dy
Therefore the component of the force along the plane is downwards and equals mg sin a.
         dU              dh
%  -- __   -.mg      -.
oz               oz
Therefore the component along the z-axis tends to move the particle normally into the plane and has a magnitude equal to mg cos a.. The components along the #-axis and the z-axis produce no motion because X equals zero and Z is exactly balanced by the reaction of the plane. The foregoing results may be verified by finding the components of mg by the common method, i.e., by taking projections of mg along the axes.
2. A rigid body which is free to rotate about a horizontal axis is displaced through an angle 6. Find the torque due to the gravitational field of the earth.